23.1.406 problem 393

Internal problem ID [5013]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 393
Date solved : Tuesday, September 30, 2025 at 11:22:57 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {-x^{2}+1}&=1+y^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 9

ode:=diff(y(x),x)*(-x^2+1)^(1/2) = 1+y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \tan \left (\arcsin \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.149 (sec). Leaf size: 42

ode=D[y[x],x]*Sqrt[1-x^2]==1+y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ][\arcsin (x)+c_1]\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 0.573 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt(1 - x**2)*Derivative(y(x), x) - y(x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \tan {\left (C_{1} + \operatorname {asin}{\left (x \right )} \right )} \]